Format guide for AIRCC - AIRCC Publishing Corporation

International Journal of Computer Networks & Communications (IJCNC) Vol.7, No.1, January 2015

A GENETIC ALGORITHM FOR CONSTRUCTING BROADCAST TREES WITH COST AND DELAY CONSTRAINTS IN COMPUTER NETWORKS Ahmed Y. Hamed1and Ghazi Al-Naymat2 1

2

College of Applied Studies and Community Services, University of Dammam, KSA Deanship of Information & Communication Technology,University of Dammam, KSA

ABSTRACT We refer to the problem of constructing broadcast trees with cost and delay constraints in the networks as a delay-constrained minimum spanning tree problem in directed networks. Hence it is necessary determining a spanning tree of minimal cost to connect the source node to all nodes subject to delay constraints on broadcast routing. In this paper, we proposed a genetic algorithm for solving broadcast routing by finding the low-cost broadcast tree with minimum cost and delay constraints. In this research we present a genetic algorithm to find the broadcast routing tree of a given network in terms of its links. The algorithm uses the connection matrix of the given network to find the spanning trees and considers the weights of the links to obtain the minimum spanning tree. Our proposed algorithm is able to find a better solution, fast convergence speed and high reliability. The scalability and the performance of the algorithm with increasing number of network nodes are also encouraging.

KEYWORDS Computer Networks, Broadcast Trees,Genetic Algorithm

1. INTRODUCTION In computer networks, broadcasting refers to transmitting a packet that will be received by every device in the network [1]. In practice, the scope of the broadcast is limited to a broadcast domain. Broadcasting a message is in contrast to unicast addressing in which a host sends datagrams to another single host identified by a unique IP address. Broadcast is a communication function that a node, called the source, sends messages to all the other nodes in the networks. Broadcast is an important function in applications of ad-hoc networks, such as in cooperative operations, group discussions, and so on. Broadcast routing is finding a broadcast tree, which is rooted from the source and contains all the nodes in the network. The cost of a broadcast is defined as the sum of cost of all the links that transmit the broadcast message in the broadcast tree. The spanning tree is defined as a connected sub-graph of that contains all its nodes and nocycles[2][3]. A simple approach is to use elementary tree-transformation which is based on the addition of a chord and deletion of an appropriate branch from a spanning tree. Thus, starting from any spanning tree of the graph with n nodes, one can generate all spanning trees by successive cyclic exchange[2]. This straightforward approach is difficult to computerize. So, DOI : 10.5121/ijcnc.2015.7103

35


Aggarwal[4] presented another algorithm, uses Cartesian products of (n-1) vertex cut sets whose elements are the branches connected to any of the (n-1) nodes of the graph. In this paper, we present a genetic algorithm to find the broadcast routing tree of a given network in terms of its links. The algorithm uses the connection matrix of a given network to find the spanning trees, and then it considers the weights of the links to obtain the minimum spanning tree. The paper is organized as follows: Section 2takes account of the related work. In Section 3, we define the problem that we are aiming to solve. Section 4 highlights some facts about the spanning tree structure. Section 5 discusses our proposed algorithm and all its steps. In Section 6, we provide our experiments that demonstrate our technique’s findings. Finally, we provide our conclusions in Section 7. The table below lists most of the used notations in this paper. Notations G N E xij d(e) c(e) M MT Ti C(T) D(T) IN Ri Ci RC RD

A network graph. The number ofnodesinG. The number oflinks inG. A link between node i and node jin G. Thedelayof a link. The cost of a link. Theconnectionmatrix ofthe given network. Theconnectionmatrixofthespanningtree. The spanningtree-i. The total cost of a spanning tree. Thedelayofaspanningtree. An initiallynumbers of the spanningtrees. A rowiin a matrix. A columniin a matrix. The require cost. The require delay.

2. RELATED WORK The minimum spanning tree problem (MSTP, Busacker[5] has been well studied in this framework, and efficient algorithms are widely known to solve this problem[6][7]. With minor changes, the same algorithms can be used to find the maximum spanning tree as well. Younes[8] presented an algorithm to find the spanning trees of the computer network in terms of its links for using it in computing the network reliability. Young[3]discussed these problems and described the first distributed algorithm for constructing minimal spanning trees. The algorithm and the principles and techniques underlying its design find application in large communication networks and large multiprocessor computer systems. Chekuri [9] focused on scheduling to minimize two related objectives: the maximum response time and maximum delay factor.

36


Fard [10] demonstrated how the spanning trees of a network can be utilized to determine a node pair across which a link should be added in order to achieve the maximum all-terminal reliability improvement in a network. Yamada [11] formulated the spanning tree detection problem, and developed heuristic as well as exact algorithms to solve the problem. With these algorithms, problems with up to 1000 nodes have been solved within a few seconds. The algorithms have been extended to list up all the spanning trees of an arbitrary specified weight. Kardan[12] presented the complete version of DBMST that regards packet loss and dynamism of the network topology and provides reliable communication. The proposed method decreases the power consumption of reliable broadcasting up to 15% in comparison with the basic version. In DBMST, network structure adapts to topology variations quickly while it takes several hours for other methods to find out variations in network. Salama [13] formulated the problem of constructing broadcast trees for real-time traffic with delay constraints in networks with asymmetric link loads as a delay-constrained minimum spanning tree (DCMST) problem in directed networks. The Delay-Constrained Capacitated Minimum Spanning Tree (DC-CMST) is a recently proposed problem which arises in the design of the topology of communications networks. The DC-CMST proposes the joint optimization of the network topology in terms of the traffic capacity and its mean time delay[14]. He presented an evolutionary algorithm which uses Dandelion-encoding is proposed to solve the problem. Younes[15] presented the broadcast routing algorithms that aim to minimize the cost of the routing path. Sameh [16]presented an algorithm to manipulate the problem of minimizing the mean completion delay in wireless broadcast for instantly decodable network coding. Salama [17] formulated the problem of constructing broadcast trees for real-time traffic with delay constraints in networks with asymmetric link loads as a delay-constrained minimum spanning tree (DCMST) problem in directed networks. They proved that this problem is NPcomplete, and they proposed an efficient heuristic to solve the problem based on Prim’s algorithm for the unconstrained minimum spanning tree problem. This is the first heuristic designed specifically for solving the DCMST problem.

3. PROBLEM DESCRIPTION Given a network in which each link has a fixed cost and delay, find a broadcast tree such that the cost of the broadcast tree is minimized and delay is constrained. Thus, we present a genetic algorithm to find the broadcast routing tree of a given network in terms of its links which will solve our problem. The algorithm uses the connection matrix of a given network to find the spanning trees, and also is based on the weights of the links to obtain the minimum spanning tree. The proposed algorithm goal is able to find a better solution, fast convergence speed and high reliability when obtaining the spanning trees. Extensive simulations have been conducted and the results have demonstrated the efficiency of the proposed algorithms The definition of the problems as follows: 37


A network is usually represented as a weighted directed graph where N denotes the set of nodes and E denotes the set of communication links connecting the nodes. and denote the number of nodes and links in the network respectively. Any link e E has a cost and a delay associated with it. and may take any positive real values. A spanning tree is rooted at a source node and contains exactly one path from s to any node

A path

–

. The total cost of a tree

is simply:

is the set of tree links connecting s to

The cost of the path

is:

And the delay of that path is:

Thus the maximum delay of a spanning tree is:

Our problem in directed networks is: constructs the spanning tree T(s) rooted at s that has minimum total cost among all possible spanning trees rooted at s which have a maximum delay less than or equal to a given delay constraint D. The problem can be expressed as: Given a directed network , a positive for each e , a source , a positive delay constraint RD and a positive cost RC, find a spanning tree T(s) that satisfy:

Where RC is the require cost.

Where RD id the require delay.

38


4. THE SPANNING TREE A tree T is considered as a spanning tree of graph G if T is a connected sub graph of G and contains all nodes of G. A link in T is called a branch of T, while a link of G that is not in T is called chord. For a connected graph of N nodes and E links, the spanning tree has (N-1) branches and (E-N+1) chords.

I.e., the characterization of the spanning tree as: Contains(N-1)edges. ContainsallnodesofG. Isa connectedsubgraphofG. The following subsections detail the above three characterizations of the spanning tree.

4.1.

TheSpanningTreeContainingallNodes

In our proposed algorithm, one of the main steps is to validate that the spanning tree contains all nodes of the given network.To illustrate that, we consider a network of 8 nodes and 13 edges as shown in figure1.

Figure 1asamplenetwork

The edges 1-2, 1-3, 1-6, 2-4, 3-5, 6-7, 7-8 of the above figure represent a spanning tree T as shown in Figure 2. The tree T contains 7 edges (N-1 edges) this satisfies the first feature of the spanning tree (it contains N-1 edges).The tree also contains all nodes of the above figure and this is the second feature. Finally,the tree forms a connected graph which means that all of its nodes are connected and this is the third feature of the spanning tree.

39


Figure 2A spanning tree of Figure 1.

The connection matrix M describes all direct connections between the nodes of a graph. The weighted connection matrix is a square matrix of dimension n x n. Entries of M are defined such that the entry Mij at the intersection of row i and column j, represents a connection from node i to node j. The rules that define a weighted connection matrix are as follows:

That is, the connection matrix of the Figure 1is:

Figure 3Theconnectionmatrixof tree inFigure 1 The spanning tree T contains all nodes of the given network, if all rows and columns ofM T are non-zero elements (MT is the connection matrix of T, which is created using Equation (7)). For example, if we consider the above spanning tree (Figure 2), the connection matrix MT is in the following form:

Figure 4Theconnectionmatrixofthe above spanningtree in Figure 2 40


In Figure 4, we notice that the matrix does not contain any row or column that all its elements are zeros. Hence, the spanning tree of Figure 2 contains all the nodes of the given network (Figure 1).

4.2. The Spanning Tree is a Connected Sub Graph. In the proposed algorithm, we prove that the spanning tree is a connected sub graph as follows: A graph G is connected if there is at least one path between every pair of nodes i and j, which minimally requires a spanning tree with edges. The sub graph is connected if all its nodes are connected. To confirm if a spanning tree T is a connected sub graph or not, we can repeat the union between the elements of both R1 and Rj in the matrix MT where Rj is a list of all indexes of the none zero elements in R1 and it. On other hand, we can use the following relation (8) which was used by Younes[8]: Do {

.

Figure 5Example to check the sub routine in Equation (8) We use the example in Figure 5 to illustrate our method in checking the connectivity of the sub graph. Below you see the steps the union operations are performed between R1 and Rj and how R1 is updated in each iteration.

41


If there exists one zero element in R1, this mean the spanning tree Ti is not aconnected sub graph. Hence, the spanning tree (1 0 1 0 1 1 1 0) is a connected sub graph.To the best of knowledge, this technique is a unique one and not previously used to validate whether the tree is connected or not.

5. THE PROPOSED GENETIC ALGORITHM (GA) In the proposed GA, each spanning tree is represented by a binary string that can be used as a chromosome. Each element of the chromosome represents a link in the network topology. So, for a network of N nodes and E edges, there are E string components in each candidate solution x. Each chromosome must contains N-1 none zero elements. For example, the spanning tree of Figure 2is represented as a chromosome as shown in Figure 6. 1-2

1-3

1-6

2-3

2-4

3-5

3-6

4-5

4-8

5-7

5-8

6-7

7-8

1

1

1

0

1

1

0

0

0

0

0

1

1

Figure 6Thechromosome form

5.1.

Initial Population

The initial population is generated according to the following steps: 1. Achromosomex intheinitialpopulationcanbe generatedas shownin Figure 6. 2. ThechromosomemustcontainonlyN-1nonezeroelements 3. Repeatsteps1 to2 to generatepop_size(population size), numberofchromosomes.

5.2.

Fitness Function

The fitness function is used in each iteration of the genetic algorithm to evaluate the quality of all proposed solutions to the problem in the current population. Hence; the fitness function is used as a filter function to eliminate all non-candidate solutions. The fitness function used to verify the solution is as follow: The chromosome must contain all nodes of the network. The chromosome must be a connected sub graph using Equation (8).

5.3.

Genetic Crossover Operation

The crossover operation is used to breed a child from two parents by one cut point. The crossover operation will perform if the crossover ratio (Pc=0.90) is verified. The cut point israndomly selected. The crossover operation is performed as follows: Randomly selecttwochromosomesfromthecurrentpopulation. Randomlyselectthecutpoint Fillthecomponentsofthechromosome a. Bytakingthecomponentsofthefirstchromosome(fromthefirst genetothecut point)andfill uptothechild. b. Also, tacking the components of the second chromosome (from the cut point+1 to the last gene) and fill up to the child. 42


The offspring generated by crossover operation is shown in Figure 7.

Parent

1

1

1

0

Cut point 1 1 0

Child

1

1

1

0

1

1

0

1

0

1

0

1

0

1

0

1

0

0

1

0

1

0

1

0

1

0

0

0

0

0

1

1

Figure 7 Example of the crossover operation

5.4.

Genetic Mutation Operation

The mutation operation is performed on bit-by-bit basis. In the proposed approach, the mutation operation will be performed if the mutation ratio (Pm) is verified. The Pm in this approach is chosen experimentally to be 0.02. The point to be mutated is selected randomly. The offspring generated by mutation is shown in Figure 8.

1

1

1

0

1

1

0

1

0

0

0

1

0

1

1

1

0

1

1

1

1

0

0

0

1

0

Figure 8 Example of the mutation operation.

5.5.

The Algorithm

The following pseudocodeillustrates the use of our assumptions and proposed functions to find the minimum spanning tree. Algorithm Find Minimum Spanning Tree Input : Set the parameters: pop_size, max_gen, RC,RD, Pm,Pc Output : Minimum Spanning Tree 1. Generatetheinitialpopulationaccording to the steps in Section5.1. 2. gen←1. 3. While(gen< =max_gen)do { 4. P ←1 5. While(P=0.90), 43


d. Mutatethe newchildaccordingtoP mparameter(Pm